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Mathematics Stack Exchange

Questions tagged [compactness]

The compactness tag is for questions about compactness and its many variants (e.g. sequential compactness, countable compactness) as well locally compact spaces; compactifications (e.g. one-point, Stone-Čech) and other topics closely related to compactness. This includes logical compactness.

Compactness is a topological property. We say that a topological space $X$ is compact if whenever we cover $X$ by a collection of open sets we can find a finite number of open sets from the collection which cover $X$. For example, $[0,1]$ is a compact subspace of $\mathbb{R}$, but $(0,1)$ and $\mathbb{R}$ are not.

We say that a space $X$ is sequentially compact if every sequence has a convergent subsequence. These properties are equivalent for metric space, although neither implies the other in general.

This tag may also be used for questions about logical compactness, such as the compactness theorem.

More information can be found on Wikipedia.

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What's the reason that exponentials are often used when requiring compact support?

What's the reason that exponentials are often used when requiring compact support? Such as $$f_j=\frac{1}{j} e^{-\frac{1}{1-(j^2x)^2}}, |x| < 1/j^2$$ $0$ otherwise. This will have a compact support. However, I would like to understand what relevance…
mavavilj
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A question about compactness

There is the question: deduce that $Y$ is a compact subset of $(X,d)$ iff the metric space $(Y,d)$ is compact. (Given that $Y$is a subset of $X$). How to show it? (I cannot find anything to show though...)
user556970
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Prove that the set $S = \{(x,y,z,w)\in \mathbb{R} : \lvert\lvert (x,y)\rvert\rvert_1 + \lvert\lvert (z,w)\rvert\rvert_{\infty} \leq 1\}$ is compact

Prove that $S = \{(x,y,z,w)\in \mathbb{R} : \lvert\lvert (x,y)\rvert\rvert_1 + \lvert\lvert (z,w)\rvert\rvert_{\infty} \leq 1\}$ is a compact set! Where : $\lvert\lvert (x,y)\rvert\rvert_1 = \lvert x \lvert + \lvert y\rvert$ $\lvert\lvert…
Collapse
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1-pt compactification of graphs of power functions

I have misgivings about the graphs of power functions below. Are they correct? Even if correct, while the slope near ∞ for n +even and near 0 for n -even are indeed infinite, the cusps are a bit hard to swallow and I am not up to doing the…
schremmer
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Let M be a complete and $A \subset M$ be totally bounded. $\Rightarrow$ cl(A) is compact.

Claim Let $M$ be a complete and $A \subset M$ be totally bounded. $\Rightarrow$ cl(A) is compact. to prove above claim, please give me any starting point. What I know about compactness is regarding finite sub-cover for every open cover of…
delog
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Spivak Calculus on Manifolds between theorems 1-3 and 1-4

The author says If $B \subset \mathbb{R}^m$ is compact and $x \in \mathbb{R}^n$, it is easy to see that $\{ x \} \times B \subset \mathbb{R}^{m+n}$ is compact. I would like to know how to prove this.
Ponzu
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How to show that closed subset of $\mathbb{R}$ is not compact if restricted to $\mathbb{Q}$

Basicly I need to show that $\mathbb{R}\cap[0,1]\cap\mathbb{Q}$ is not compact. I was looking at some posts on this topic and all, that I found, used the finite subcover definition of compact set. I wonder if it could be done this way: A compact set…
Mykolas
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Quick question on compact support

I am reading something on Sobolev spaces. We define $D(\Omega)$ to be the set of function in $C^{\infty}(\Omega)$ that has compact support in $\Omega$. I know that compact support is completion of the set where a function is non-zero. but, what…
Lost1
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Is $\prod_{n=1}^{\infty} A_{n}$ with product topology, where $A_{n}$=${\{0,1\}}$ has discrete topology for , $n = 1,2,3,\cdots.$ a compact set?

Is $\prod_{n=1}^{\infty} A_{n}$ with product topology, where $A_{n}$=${\{0,1\}}$ has discrete topology for , $n = 1,2,3,\cdots.$ a compact set? How to show it? My approach: To show that a set is compact, we need to show that every open cover has…
A. Khan
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Compactness in weak topology

Consider $V$ a normed vector space, and equip its topological dual $V^*$ with the usual operator norm. Consider also the weak* topology on $V^*$ defined by the seminorms $(p_A)_{A\subset V \text{finite}}$ defined as follows : $\forall f \in V \quad…
James Well
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Inverse that isn't Sequentially Compact

I am looking for a function $f: R\rightarrow R$ and a sequentially compact $K\subset R$ such that the inverse $f^{-1}(K)$ is not sequentially compact. I decided to choose $f(x) = sin(x)$, but I'm not sure what $K$ could be such that $f^{-1}(K)$ is…
samsonite
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Baby Rudin Theorem 2.41 boundedness

I'm currently reading the POMA of Rudin and I don't understand the proof of boundedness of the theorem 2.41 (p. 40) of the book. It wants to prove that if $E \subset \mathbb{R}^k$ we have that every infinite subset of $E$ has a limit point in $E$…
yarmenti
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Closure of set of vectors with norm 1 in $\mathbb{C}^n$

In the proof for existence of SVD, it always says - Due to compactness, we can always find a vector $v_{1} \in \mathbb{C}^n$ such that $A\,v_{1} = \sigma_{1} \, u_{1}$. Another post explained what is meant in the proof when they say compactness.…
astudent
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Compact Sets in $\mathbb{R^{n^2}}$

I have a question of multivariable analysis and I don't know how to resolve this. The $n \times n$ orthogonal matrices form a compact subset of $\mathbb{R^{n^2}}$? I will be very grateful for the help.
Ysaac
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Show (0,1) is not compact

Let $I_n=\left(\frac{1}{n},1\right)$. Show that $(0,1)$ is not compact: show that any finite collection of $\{I_n\}$ will not cover $(0,1)$. Give me a hint.
user222887
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